Sine, Cosine and Tangent

Opposite & adjacent sides and SOHCAHTOA of angles

This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides practice problems on identifying the sides that are opposite and adjacent to a given angle.

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Related: sohcahtoa home | Finding Sine,cosine, tangent ratios | images
Sine Cosine Calculator | sohcahtoa worksheet

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sine of an angle is always the ratio of the (opposite side)/(hypotenuse).

(opposite side)/(hypotenuse)

Range of Values of Sine
For those comfortable in "Math Speak", the domain and range of Sine is as follows

Domain of Sine = all real numbers
Range of Sine = {-1 ≤ y ≤ 1}

The sine of an angle has a range of values from -1 to 1 inclusive. Below is a table of values illustrating some key sine values that span the entire range of values

Angle	Sine of the Angle
270°	sin ( 270°) = -1 (smallest value that sine can ever have)
330°	sin (330° ) = -½
0°	sin(0°) = 0
30°	sin(30°) = ½
90°	sin(90°) = 1 (greatest value that sine can ever have)

The cosine of an angle is always the ratio of the (adjacent side/ hypotenuse)

(adjacent side)/(hypotenuse)

Range of Values of Cosine
For those comfortable in "Math Speak", the domain and range of cosine is as follows

Domain of Cosine = all real numbers
Range of Cosine = {-1 ≤ y ≤ 1}

The cosine of an angle has a range of values from -1 to 1 inclusive. Below is a table of values illustrating some key cosine values that span the entire range of values

Angle	Sine of the Angle
0°	cos (0°) = 1 ( greatest value that cosine can ever have)
60°	cos (60° ) =½
90°	cos(90°) = 0
120°	cos(120°) = -½
180°	cos(180°) = -1 (smallest value that sine can ever have)

The tangent of an angle is always the ratio of the (opposite side)/(adjacent side)

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Related: sohcahtoa home | Finding Sine,cosine, tangent ratios | images
Sine Cosine Calculator | sohcahtoa worksheet

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Practice Identifying Adjacent and Opposite Sides

Identify the opposite and adjacent sides of YZX. Answer The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle--which in this case is the shaded angle in the picture.

Identify the side that is opposite of IHU and the side that is adjacent to IHU. Answer

In the triangles below, identify the hypotenuse and the sides that are oppostie and adjacent to the shaded angle.

Show Side Labels Hypotenuse = C Side opposite of x = B Side adjacent to x = A

Show Side Labels Hypotenuse = K Side opposite of Y = J Side adjacent to Y = L

Show Side Labels Hypotenuse = X Side opposite of B = Z Side adjacent to B = Y

Show Side Labels Hypotenuse = I Side opposite of A = H Side adjacent to A = J

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Related: sohcahtoa home | Finding Sine,cosine, tangent ratios
Sine Cosine Calculator | sohcahtoa worksheet

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